Download Example 6.1 The Conical Pendulum A small ball of mass m is

Example 6.1 The Conical Pendulum
A small ball of mass m is suspended from a string of length L. The ball revolves with
constant speed υ in a horizontal circle of radius r as shown in Figure 6.3. (Because the
string sweeps out the surface of a cone, the system is known as a conical pendulum.)
Find an expression for υ in terms of the geometry in Figure 6.3.
SOLUTION
Conceptualize Imagine the motion of the ball in Figure 6.3a and convince yourself
that the string sweeps out a cone and that the ball moves in a horizontal circle.
Categorize The ball in Figure 6.3 does not accelerate vertically. Therefore, we model
it as a particle in equilibrium in the vertical direction. It experiences a centripetal
acceleration in the horizontal direction, so it is modeled as a particle in uniform
circular motion in this direction.
Figure 6.3 (Example 6.1) (a) A conical pendulum. The path of the ball is a horizontal
circle. (b) The forces acting on the ball.
Analyze Let θ represent the angle between the string and the vertical. In the diagram
of forces acting on the ball in Figure 6.3b, the force T exerted by the string on the
ball is resolved into a vertical component T cos θ and a horizontal component T sin θ
acting toward the center of the circular path.
Apply the particle in equilibrium model
∑Fy = T cos θ − mg = 0
in the vertical direction:
Use Equation 6.1 from the particle in
uniform circular motion model in the
(1) T cos θ = mg
(2)
F
x
 T sin   mac 
m 2
r
horizontal direction:
Divide Equation (2) by Equation (1) and
use sin θ/cos θ = tan θ:
tan  
2
rg
Solve for υ:
  rg tan 
Incorporate r = L sin θ from the geometry
  Lg sin  tan 
in Figure 6.3a:
Finalize Notice that the speed is independent of the mass of the ball. Consider what
happens when θ goes to 90° so that the string is horizontal. Because the tangent of 90°
is infinite, the speed υ is infinite, which tells us the string cannot possibly be
horizontal. If it were, there would be no vertical component of the force T to balance
the gravitational force on the ball. That is why we mentioned in regard to Figure 6.1
that the puck’s weight in the figure is supported by a frictionless table.
Example 6.2 How Fast Can It Spin?
A puck of mass 0.500 kg is attached to the end of a cord 1.50 m long. The puck
moves in a horizontal circle as shown in Figure 6.1. If the cord can withstand a
maximum tension of 50.0 N, what is the maximum speed at which the puck can move
before the cord breaks? Assume the string remains horizontal during the motion.
SOLUTION
Conceptualize It makes sense that the stronger the cord, the faster the puck can move
before the cord breaks. Also, we expect a more massive puck to break the cord at a
lower speed. (Imagine whirling a bowling ball on the cord!)
Categorize Because the puck moves in a circular path, we model it as a particle in
uniform circular motion.
Analyze Incorporate the tension
T m
2
and the centripetal acceleration
r
into Newton’s second law as
described by Equation 6.1:
Solve for υ:
Find the maximum speed the puck
can have, which corresponds to the
Tr
m
(1)  
max 
Tmax r

m
 50.0N 1.50m   12.2 m s
0.500kg
maximum tension the string can
withstand:
Finalize Equation (1) shows that υ increases with T and decreases with larger m, as
we expected from our conceptualization of the problem.
WHAT IF? Suppose the puck moves in a circle of larger radius at the same speed υ.
Is the cord more likely or less likely to break?
Answer The larger radius means that the change in the direction of the velocity vector
will be smaller in a given time interval. Therefore, the acceleration is smaller and the
required tension in the string is smaller. As a result, the string is less likely to break
when the puck travels in a circle of larger radius.
Example 6.3 What Is the Maximum Speed of the Car?
A 1 500-kg car moving on a flat, horizontal road negotiates a curve as shown in
Figure 6.4a. If the radius of the curve is 35.0 m and the coefficient of static friction
between the tires and dry pavement is 0.523, find the maximum speed the car can
have and still make the turn successfully.
SOLUTION
Conceptualize Imagine that the curved roadway is part of a large circle so that the car
is moving in a circular path.
Categorize Based on the Conceptualize step of the problem, we model the car as a
particle in uniform circular motion in the horizontal direction. The car is not
accelerating vertically, so it is modeled as a particle in equilibrium in the vertical
direction.
Analyze Figure 6.4b shows the forces on the car. The force that enables the car to
remain in its circular path is the force of static friction. (It is static because no slipping
occurs at the point of contact between road and tires. If this force of static friction
were zero—for example, if the car were on an icy road—the car would continue in a
straight line and slide off the curved road.) The maximum speed υmax the car can have
around the curve is the speed at which it is on the verge of skidding outward. At this
point, the friction force has its maximum value f s,max = μsn.
Figure 6.4 (Example 6.3) (a) The force of static friction directed toward the center of
the curve keeps the car moving in a circular path. (b) The forces acting on the car.
Apply Equation 6.1 from the
(1) f s ,max  s n  m
2
max
particle in uniform circular
r
motion model in the radial
direction for the maximum
speed condition:
Apply the particle in
∑Fy = 0 → n − mg = 0 → n = mg
equilibrium model to the car in
the vertical direction:
Solve Equation (1) for the
(2)
maximum speed and substitute
max 
 s nr
m

 s mgr
m
  s gr
for n:
Substitute numerical values:
max 
 0.523  9.80 m s 2   35.0m   13.4 m s
Finalize This speed is equivalent to 30.0 mi/h. Therefore, if the speed limit on this
roadway is higher than 30 mi/h, this roadway could benefit greatly from some
banking, as in the next example! Notice that the maximum speed does not depend on
the mass of the car, which is why curved highways do not need multiple speed limits
to cover the various masses of vehicles using the road.
WHAT IF? Suppose a car travels this curve on a wet day and begins to skid on the
curve when its speed reaches only 8.00 m/s. What can we say about the coefficient of
static friction in this case?
Answer The coefficient of static friction between the tires and a wet road should be
smaller than that between the tires and a dry road. This expectation is consistent with
experience with driving because a skid is more likely on a wet road than a dry road.
To check our suspicion, we can solve Equation (2) for the coefficient of static
friction:
s 
2
 max
gr
Substituting the numerical values gives
s 
2
max
gr
8.00 m s 

 0.187
9.80 m s2  35.0m
2
which is indeed smaller than the coefficient of 0.523 for the dry road.
Example 6.4 The Banked Roadway
A civil engineer wishes to redesign the curved roadway in Example 6.3 in such a way
that a car will not have to rely on friction to round the curve without skidding. In other
words, a car moving at the designated speed can negotiate the curve even when the
road is covered with ice. Such a road is usually banked, which means that the
roadway is tilted toward the inside of the curve as seen in the opening photograph for
this chapter. Suppose the designated speed for the road is to be 13.4 m/s (30.0 mi/h)
and the radius of the curve is 35.0 m. At what angle should the curve be banked?
SOLUTION
Conceptualize The difference between this example and Example 6.3 is that the car is
no longer moving on a flat roadway. Figure 6.5 shows the banked roadway, with the
center of the circular path of the car far to the left of the figure. Notice that the
horizontal component of the normal force participates in causing the car’s centripetal
acceleration.
Categorize As in Example 6.3, the car is modeled as a particle in equilibrium in the
vertical direction and a particle in uniform circular motion in the horizontal direction.
Figure 6.5 (Example 6.4) A car moves into the page and is rounding a curve on a
road banked at an angle θ to the horizontal. When friction is neglected, the force that
causes the centripetal acceleration and keeps the car moving in its circular path is the
horizontal component of the normal force.
Analyze On a level (unbanked) road, the force that causes the centripetal acceleration
is the force of static friction between tires and the road as we saw in the preceding
example. If the road is banked at an angle θ as in Figure 6.5, however, the normal
force n has a horizontal component toward the center of the curve. Because the road
is to be designed so that the force of static friction is zero, the component nx = n sin θ
is the only force that causes the centripetal acceleration.
Write Newton’s second law for the car in
(1)
 Fr  n sin  
the radial direction, which is the x
m 2
r
direction:
Apply the particle in equilibrium model
∑Fy = n cos θ − mg = 0
to the car in the vertical direction:
(2) n cos θ = mg
Divide Equation (1) by Equation (2):
tan  
(3)
2
rg
Solve for the angle θ:
2


13.4 m s 

  27.6
  tan 
  35.0m  9.80 m s 2 


1


Finalize Equation (3) shows that the banking angle is independent of the mass of the
vehicle negotiating the curve. If a car rounds the curve at a speed less than 13.4 m/s,
the centripetal acceleration decreases. Therefore, the normal force, which is
unchanged, is sufficient to cause two accelerations: the lower centripetal acceleration
and an acceleration of the car down the inclined roadway. Consequently, an additional
friction force parallel to the roadway and upward is needed to keep the car from
sliding down the bank (to the left in Fig. 6.5). Similarly, a driver attempting to
negotiate the curve at a speed greater than 13.4 m/s has to depend on friction to keep
from sliding up the bank (to the right in Fig. 6.5).
WHAT IF? Imagine that this same roadway were built on Mars in the future to
connect different colony centers. Could it be traveled at the same speed?
Answer The reduced gravitational force on Mars would mean that the car is not
pressed as tightly to the roadway. The reduced normal force results in a smaller
component of the normal force toward the center of the circle. This smaller
component would not be sufficient to provide the centripetal acceleration associated
with the original speed. The centripetal acceleration must be reduced, which can be
done by reducing the speed υ.
Mathematically, notice that Equation (3) shows that the speed υ is proportional to
the square root of g for a roadway of fixed radius r banked at a fixed angle θ.
Therefore, if g is smaller, as it is on Mars, the speed υ with which the roadway can be
safely traveled is also smaller.
Example 6.5 Riding the Ferris Wheel
A child of mass m rides on a Ferris wheel as shown in Figure 6.6a. The child moves in
a vertical circle of radius 10.0 m at a constant speed of 3.00 m/s.
(A) Determine the force exerted by the seat on the child at the bottom of the ride.
Express your answer in terms of the weight of the child, mg.
SOLUTION
Conceptualize Look carefully at Figure 6.6a. Based on experiences you may have
had on a Ferris wheel or driving over small hills on a roadway, you would expect to
feel lighter at the top of the path. Similarly, you would expect to feel heavier at the
bottom of the path. At both the bottom of the path and the top, the normal and
gravitational forces on the child act in opposite directions. The vector sum of these
two forces gives a force of constant magnitude that keeps the child moving in a
circular path at a constant speed. To yield net force vectors with the same magnitude,
the normal force at the bottom must be greater than that at the top.
Figure 6.6 (Example 6.5) (a) A child rides on a Ferris wheel. (b) The forces acting on
the child at the bottom of the path. (c) The forces acting on the child at the top of the
path.
Categorize Because the speed of the child is constant, we can categorize this problem
as one involving a particle (the child) in uniform circular motion, complicated by the
gravitational force acting at all times on the child.
Analyze We draw a diagram of forces acting on the child at the bottom of the ride as
shown in Figure 6.6b. The only forces acting on him are the downward gravitational
force F g  mg and the upward force n bot exerted by the seat. The net upward force
on the child that provides his centripetal acceleration has a magnitude nbot − mg.
Using the particle in uniform circular
motion model, apply Newton’s second
 F  nbot  mg  m
2
r
law to the child in the radial direction
when he is at the bottom of the ride:
Solve for the force exerted by the seat
nbot
on the child:
Substitute numerical values given for
the speed and radius:
2
 2 
 mg  m  mg 1  
r
 rg 
2


3.00 m s 



nbot  mg 1 
 10.0m  9.80 m s 2 


 1.09mg


Hence, the magnitude of the force n bot exerted by the seat on the child is greater than
the weight of the child by a factor of 1.09. So, the child experiences an apparent
weight that is greater than his true weight by a factor of 1.09.
(B) Determine the force exerted by the seat on the child at the top of the ride.
SOLUTION
Analyze The diagram of forces acting on the child at the top of the ride is shown in
Figure 6.6c. The net downward force that provides the centripetal acceleration has a
magnitude mg − ntop .
Apply Newton’s second law to the
 F  mg  ntop  m
child at this position:
Solve for the force exerted by the seat
on the child:
ntop  mg  m
2
2
r
 2 
 mg 1  
r
 rg 
Substitute numerical values:
2


3.00 m s 


ntop  mg 1 
 10.0m  9.80 m s 2 


 0.908mg


In this case, the magnitude of the force exerted by the seat on the child is less than his
true weight by a factor of 0.908, and the child feels lighter.
Finalize The variations in the normal force are consistent with our prediction in the
Conceptualize step of the problem.
WHAT IF? Suppose a defect in the Ferris wheel mechanism causes the speed of the
child to increase to 10.0 m/s. What does the child experience at the top of the ride in
this case?
Answer If the calculation above is performed with υ = 10.0 m/s, the magnitude of the
normal force at the top of the ride is negative, which is impossible. We interpret it to
mean that the required centripetal acceleration of the child is larger than that due to
gravity. As a result, the child will lose contact with the seat and will only stay in his
circular path if there is a safety bar or a seat belt that provides a downward force on
him to keep him in his seat. At the bottom of the ride, the normal force is 2.02 mg,
which would be uncomfortable.
Example 6.6 Keep Your Eye on the Ball
A small sphere of mass m is attached to the end of a cord of length R and set into
motion in a vertical circle about a fixed point O as illustrated in Figure 6.9. Determine
the tangential acceleration of the sphere and the tension in the cord at any instant
when the speed of the sphere is υ and the cord makes an angle θ with the vertical.
SOLUTION
Conceptualize Compare the motion of the sphere in Figure 6.9 with that of the child
in Figure 6.6a associated with Example 6.5. Both objects travel in a circular path.
Unlike the child in Example 6.5, however, the speed of the sphere is not uniform in
this example because, at most points along the path, a tangential component of
acceleration arises from the gravitational force exerted on the sphere.
Categorize We model the sphere as a particle under a net force and moving in a
circular path, but it is not a particle in uniform circular motion. We need to use the
techniques discussed in this section on nonuniform circular motion.
Figure 6.9 (Example 6.6) The forces acting on a sphere of mass m connected to a
cord of length R and rotating in a vertical circle centered at O. Forces acting on the
sphere are shown when the sphere is at the top and bottom of the circle and at an
arbitrary location.
Analyze From the force diagram in Figure 6.9, we see that the only forces acting on
the sphere are the gravitational force F g  mg exerted by the Earth and the force T
exerted by the cord. We resolve F g into a tangential component mg sin θ and a radial
component mg cos θ.
From the particle under a net force
∑Ft = mg sin θ = mat
model, apply Newton’s second law to the
at = g sin θ
sphere in the tangential direction:
Apply Newton’s second law to the forces
F
r
acting on the sphere in the radial
direction, noting that both T and a r are
 T  mg cos  
m 2
R
 2

T  mg 
 cos  
 Rg

directed toward O. As noted in Section
4.5, we can use Equation 4.14 for the
centripetal acceleration of a particle even
when it moves in a circular path in
nonuniform motion:
Finalize Let us evaluate this result at the top and bottom of the circular path (Fig. 6.9):
2
 top

Ttop  mg 
 1
 Rg



2

Tbot  mg  bot  1
 Rg

These results have similar mathematical forms as those for the normal forces ntop and
nbot on the child in Example 6.5, which is consistent with the normal force on the
child playing a similar physical role in Example 6.5 as the tension in the string plays
in this example. Keep in mind, however, that the normal force n on the child in
Example 6.5 is always upward, whereas the force T in this example changes
direction because it must always point inward along the string. Also note that υ in the
expressions above varies for different positions of the sphere, as indicated by the
subscripts, whereas υ in Example 6.5 is constant.
WHAT IF? What if the ball is set in motion with a slower speed?
(A) What speed would the ball have as it passes over the top of the circle if the
tension in the cord goes to zero instantaneously at this point?
Answer Let us set the tension equal to zero in the expression for Ttop :
2
 top

0  mg 
 1  top  gR
 Rg



(B) What if the ball is set in motion such that the speed at the top is less than this
value? What happens?
Answer In this case, the ball never reaches the top of the circle. At some point on the
way up, the tension in the string goes to zero and the ball becomes a projectile. It
follows a segment of a parabolic path over the top of its motion, rejoining the circular
path on the other side when the tension becomes nonzero again.
Example 6.7 Fictitious Forces in Linear Motion
A small sphere of mass m hangs by a cord from the ceiling of a boxcar that is
accelerating to the right as shown in Figure 6.12. Both the inertial observer on the
ground in Figure 6.12a and the noninertial observer on the train in Figure 6.12b agree
that the cord makes an angle θ with respect to the vertical. The noninertial observer
claims that a force, which we know to be fictitious, causes the observed deviation of
the cord from the vertical. How is the magnitude of this force related to the boxcar’s
acceleration measured by the inertial observer in Figure 6.12a?
SOLUTION
Conceptualize Place yourself in the role of each of the two observers in Figure 6.12.
As the inertial observer on the ground, you see the boxcar accelerating and know that
the deviation of the cord is due to this acceleration. As the noninertial observer on the
boxcar, imagine that you ignore any effects of the car’s motion so that you are not
aware of its acceleration. Because you are unaware of this acceleration, you claim that
a force is pushing sideways on the sphere to cause the deviation of the cord from the
vertical. To make the conceptualization more real, try running from rest while holding
a hanging object on a string and notice that the string is at an angle to the vertical
while you are accelerating, as if a force is pushing the object backward.
Figure 6.12 (Example 6.7) A small sphere suspended from the ceiling of a boxcar
accelerating to the right is deflected as shown.
Categorize For the inertial observer, we model the sphere as a particle under a net
force in the horizontal direction and a particle in equilibrium in the vertical direction.
For the noninertial observer, the sphere is modeled as a particle in equilibrium in both
directions.
Analyze According to the inertial observer at rest (Fig. 6.12a), the forces on the
sphere are the force T exerted by the cord and the gravitational force. The inertial
observer concludes that the sphere’s acceleration is the same as that of the boxcar and
that this acceleration is provided by the horizontal component of T .
For this observer, apply the
Inertial observer
particle under a net force and

1


 2 
F
F
x
 T sin   ma
y
 T cos   mg  0
particle in equilibrium models:
According to the noninertial observer riding in the car (Fig. 6.12b), the cord also
makes an angle θ with the vertical; to that observer, however, the sphere is at rest and
so its acceleration is zero. Therefore, the noninertial observer introduces a force
(which we know to be fictitious) in the horizontal direction to balance the horizontal
component of T and claims that the net force on the sphere is zero.
Apply the particle in
equilibrium model for this

 Fx  T sin   Ffictitious  0
Noninertial observer 

 Fy  T cos   mg  0
observer in both directions:
These expressions are equivalent to Equations (1) and (2) if Ffictitious = ma, where a is
the acceleration according to the inertial observer.
Finalize If we make this substitution in the equation for  Fx above, we obtain the
same mathematical results as the inertial observer. The physical interpretation of the
cord’s deflection, however, differs in the two frames of reference.
WHAT IF? Suppose the inertial observer wants to measure the acceleration of the
train by means of the pendulum (the sphere hanging from the cord). How could she do
so?
Answer Our intuition tells us that the angle θ the cord makes with the vertical should
increase as the acceleration increases. By solving Equations (1) and (2)
simultaneously for a, we find that a = g tan θ. Therefore, the inertial observer can
determine the magnitude of the car’s acceleration by measuring the angle θ and using
that relationship. Because the deflection of the cord from the vertical serves as a
measure of acceleration, a simple pendulum can be used as an accelerometer.
Example 6.8 Sphere Falling in Oil
A small sphere of mass 2.00 g is released from rest in a large vessel filled with oil,
where it experiences a resistive force proportional to its speed. The sphere reaches a
terminal speed of 5.00 cm/s. Determine the time constant τ and the time at which the
sphere reaches 90.0% of its terminal speed.
SOLUTION
Conceptualize With the help of Figure 6.13, imagine dropping the sphere into the oil
and watching it sink to the bottom of the vessel. If you have some thick shampoo in a
clear container, drop a marble in it and observe the motion of the marble.
Categorize We model the sphere as a particle under a net force, with one of the
forces being a resistive force that depends on the speed of the sphere. This model
leads to the result in Equation 6.5.
Analyze From Equation 6.5,
b
mg

  
m
 m    
b
 mg  g

5.00 cm s
 5.10 103 s
980 cm s 2
evaluate the coefficient b:
Evaluate the time constant τ:
Substitute numerical values:
T
Find the time t at which the
0.900T  T 1  e  t  
sphere reaches a speed of
1  e  t   0.900
0.900 υT by setting υ = 0.900
υT in Equation 6.6 and solving
for t:
e  t   0.100

t

 ln  0.100   2.30
t  2.30  2.30  5.10  10 3 s   11.7  10 3 s
=11.7ms
Finalize The sphere reaches 90.0% of its terminal speed in a very short time interval.
You should have also seen this behavior if you performed the activity with the marble
and the shampoo. Because of the short time interval required to reach terminal
velocity, you may not have noticed the time interval at all. The marble may have
appeared to immediately begin moving through the shampoo at a constant velocity.
Conceptual Example 6.9 The Skysurfer
Consider a skysurfer (Fig. 6.15) who jumps from a plane with his feet attached firmly
to his surfboard, does some tricks, and then opens his parachute. Describe the forces
acting on him during these maneuvers.
SOLUTION
When the surfer first steps out of the plane, he has no vertical velocity. The downward
gravitational force causes him to accelerate toward the ground. As his downward
speed increases, so does the upward resistive force exerted by the air on his body and
the board. This upward force reduces their acceleration, and so their speed increases
more slowly. Eventually, they are going so fast that the upward resistive force
matches the downward gravitational force. Now the net force is zero and they no
longer accelerate, but instead reach their terminal speed. At some point after reaching
terminal speed, he opens his parachute, resulting in a drastic increase in the upward
resistive force. The net force (and therefore the acceleration) is now upward, in the
direction opposite the direction of the velocity. The downward velocity therefore
decreases rapidly, and the resistive force on the parachute also decreases. Eventually,
the upward resistive force and the downward gravitational force balance each other
again and a much smaller terminal speed is reached, permitting a safe landing.
Figure 6.15 (Conceptual Example 6.9) A skysurfer.
Oliver Furrer/Jupiter Images
(Contrary to popular belief, the velocity vector of a skydiver never points upward.
You may have seen a video in which a skydiver appears to “rocket” upward once the
parachute opens. In fact, what happens is that the skydiver slows down but the person
holding the camera continues falling at high speed.)
Example 6.10 Falling Coffee Filters
The dependence of resistive force on the square of the speed is a simplification model.
Let’s test the model for a specific situation. Imagine an experiment in which we drop
a series of bowl-shaped, pleated coffee filters and measure their terminal speeds.
Table 6.2 on page 166 presents typical terminal speed data from a real experiment
using these coffee filters as they fall through the air. The time constant τ is small, so a
dropped filter quickly reaches terminal speed. Each filter has a mass of 1.64 g. When
the filters are nested together, they combine in such a way that the front-facing
surface area does not increase. Determine the relationship between the resistive force
exerted by the air and the speed of the falling filters.
SOLUTION
Conceptualize Imagine dropping the coffee filters through the air. (If you have some
coffee filters, try dropping them.) Because of the relatively small mass of the coffee
filter, you probably won’t notice the time interval during which there is an
acceleration. The filters will appear to fall at constant velocity immediately upon
leaving your hand.
Categorize Because a filter moves at constant velocity, we model it as a particle in
equilibrium.
Analyze At terminal speed, the upward resistive force on the filter balances the
downward gravitational force so that R = mg.
Evaluate the magnitude of
 1kg 
2
R  mg  1.60g  
 9.80 m s  0.0161N
 1000g 

the resistive force:

Likewise, two filters nested together experience 0.032 2 N of resistive force, and so
forth. These values of resistive force are shown in the far right column of Table 6.2. A
graph of the resistive force on the filters as a function of terminal speed is shown in
Figure 6.16a. A straight line is not a good fit, indicating that the resistive force is not
proportional to the speed. The behavior is more clearly seen in Figure 6.16b, in which
the resistive force is plotted as a function of the square of the terminal speed. This
graph indicates that the resistive force is proportional to the square of the speed as
suggested by Equation 6.7.
Table 6.2 Terminal Speed and Resistive Force for Nested Coffee Filters
Number of
Filters
υT (m/s)a
R (N)
1
1.01
0.016 1
2
1.40
0.032 2
3
1.63
0.048 3
4
2.00
0.064 4
5
2.25
0.080 5
6
2.40
0.096 6
7
2.57
0.112 7
8
2.80
0.128 8
9
3.05
0.144 9
10
3.22
0.161 0
aAll
values of υT are approximate.
Finalize Here is a good opportunity for you to take some actual data at home on real
coffee filters and see if you can reproduce the results shown in Figure 6.16. If you
have shampoo and a marble as mentioned in Example 6.8, take data on that system
too and see if the resistive force is appropriately modeled as being proportional to the
speed.
Figure 6.16 (Example 6.10) (a) Relationship between the resistive force acting on
falling coffee filters and their terminal speed. (b) Graph relating the resistive force to
the square of the terminal speed.
Example 6.11 Resistive Force Exerted on a Baseball
A pitcher hurls a 0.145-kg baseball past a batter at 40.2 m/s (= 90 mi/h). Find the
resistive force acting on the ball at this speed.
SOLUTION
Conceptualize This example is different from the previous ones in that the object is
now moving horizontally through the air instead of moving vertically under the
influence of gravity and the resistive force. The resistive force causes the ball to slow
down, and gravity causes its trajectory to curve downward. We simplify the situation
by assuming the velocity vector is exactly horizontal at the instant it is traveling at
40.2 m/s.
Categorize In general, the ball is a particle under a net force. Because we are
considering only one instant of time, however, we are not concerned about
acceleration, so the problem involves only finding the value of one of the forces.
Analyze To determine the
drag coefficient D, imagine
D
2mg
T2  A
that we drop the baseball and
allow it to reach terminal
speed. Solve Equation 6.10 for
D:
Use this expression for D in
 
1  2mg 
2
R  12 D  A   2
  A  mg  
2    A 
  
2
2
Equation 6.7 to find an
expression for the magnitude
of the resistive force:
Substitute numerical values,
using the terminal speed from
2
 40.2 m s 
R   0.145 kg  9.80 m s 
  1.2 N
 43m s 

2

Table 6.1:
Finalize The magnitude of the resistive force is similar in magnitude to the weight of
the baseball, which is about 1.4 N. Therefore, air resistance plays a major role in the
motion of the ball, as evidenced by the variety of curve balls, floaters, sinkers, and the
like thrown by baseball pitchers.
Objective Questions 1. denotes answer available in Student Solutions Manual/Study
Guide
1. A child is practicing for a BMX race. His speed remains constant as he goes
counterclockwise around a level track with two straight sections and two nearly
semicircular sections as shown in the aerial view of Figure OQ6.1. (a) Rank the
magnitudes of his acceleration at the points A, B, C, D, and E from largest to
smallest. If his acceleration is the same size at two points, display that fact in your
ranking. If his acceleration is zero, display that fact. (b) What are the directions of
his velocity at points A, B, and C? For each point, choose one: north, south, east,
west, or nonexistent. (c) What are the directions of his acceleration at points A, B,
and C?
Figure OQ6.1.
2. Consider a skydiver who has stepped from a helicopter and is falling through air.
Before she reaches terminal speed and long before she opens her parachute, does
her speed (a) increase, (b) decrease, or (c) stay constant?
3. A door in a hospital has a pneumatic closer that pulls the door shut such that the
doorknob moves with constant speed over most of its path. In this part of its
motion, (a) does the doorknob experience a centripetal acceleration? (b) Does it
experience a tangential acceleration?
4. A pendulum consists of a small object called a bob hanging from a light cord of
fixed length, with the top end of the cord fixed, as represented in Figure OQ6.4.
The bob moves without friction, swinging equally high on both sides. It moves
from its turning point A through point B and reaches its maximum speed at point
C. (a) Of these points, is there a point where the bob has nonzero radial
acceleration and zero tangential acceleration? If so, which point? What is the
direction of its total acceleration at this point? (b) Of these points, is there a point
where the bob has nonzero tangential acceleration and zero radial acceleration? If
so, which point? What is the direction of its total acceleration at this point? (c) Is
there a point where the bob has no acceleration? If so, which point? (d) Is there a
point where the bob has both nonzero tangential and radial acceleration? If so,
which point? What is the direction of its total acceleration at this point?
Figure OQ6.4
5. As a raindrop falls through the atmosphere, its speed initially changes as it falls
toward the Earth. Before the raindrop reaches its terminal speed, does the
magnitude of its acceleration (a) increase, (b) decrease, (c) stay constant at zero,
(d) stay constant at 9.80 m/s2 , or (e) stay constant at some other value?
6. An office door is given a sharp push and swings open against a pneumatic device
that slows the door down and then reverses its motion. At the moment the door is
open the widest, (a) does the doorknob have a centripetal acceleration? (b) Does it
have a tangential acceleration?
7. Before takeoff on an airplane, an inquisitive student on the plane dangles an iPod
by its earphone wire. It hangs straight down as the plane is at rest waiting to take
off. The plane then gains speed rapidly as it moves down the runway. (i) Relative
to the student's hand, does the iPod (a) shift toward the front of the plane, (b)
continue to hang straight down, or (c) shift toward the back of the plane? (ii) The
speed of the plane increases at a constant rate over a time interval of several
seconds. During this interval, does the angle the earphone wire makes with the
vertical (a) increase, (b) stay constant, or (c) decrease?
Conceptual Questions 1. denotes answer available in Student Solutions
Manual/Study Guide
1. What forces cause (a) an automobile, (b) a propeller-driven airplane, and (c) a
rowboat to move?
2. A falling skydiver reaches terminal speed with her parachute closed. After the
parachute is opened, what parameters change to decrease this terminal speed?
3. An object executes circular motion with constant speed whenever a net force of
constant magnitude acts perpendicular to the velocity. What happens to the speed
if the force is not perpendicular to the velocity?
4. Describe the path of a moving body in the event that (a) its acceleration is
constant in magnitude at all times and perpendicular to the velocity, and (b) its
acceleration is constant in magnitude at all times and parallel to the velocity.
5. The observer in the accelerating elevator of Example 5.8 would claim that the
“weight” of the fish is T, the scale reading, but this answer is obviously wrong.
Why does this observation differ from that of a person outside the elevator, at rest
with respect to the Earth?
6. If someone told you that astronauts are weightless in orbit because they are
beyond the pull of gravity, would you accept the statement? Explain.
7. It has been suggested that rotating cylinders about 20 km in length and 8 km in
diameter be placed in space and used as colonies. The purpose of the rotation is to
simulate gravity for the inhabitants. Explain this concept for producing an
effective imitation of gravity.
8. Consider a small raindrop and a large raindrop falling through the atmosphere. (a)
Compare their terminal speeds. (b) What are their accelerations when they reach
terminal speed?
9. Why does a pilot tend to black out when pulling out of a steep dive?
10. A pail of water can be whirled in a vertical path such that no water is spilled. Why
does the water stay in the pail, even when the pail is above your head?
11. “If the current position and velocity of every particle in the Universe were known,
together with the laws describing the forces that particles exert on one another, the
whole future of the Universe could be calculated. The future is determinate and
preordained. Free will is an illusion.” Do you agree with this thesis? Argue for or
against it.
Problems
The problems found in this chapter may be assigned online in Enhanced
WebAssign
1. straightforward; 2. intermediate; 3. challenging
1. full solution available in the Studen Solutions Manual/Study Guide
Analysis Model tutorial available in Enhanced WebAssign
Guided Problem
Master It tutorial available in Enhanced WebAssign
Watch It video solution available in Enhanced WebAssign
Section 6.1 Extending the Particle in Uniform Circular Motion Model
1. A light string can support a stationary hanging load of 25.0 kg before breaking.
An object of mass m = 3.00 kg attached to the string rotates on a frictionless,
horizontal table in a circle of radius r = 0.800 m, and the other end of the string is
held fixed as in Figure P6.1. What range of speeds can the object have before the
string breaks?
Figure P6.1
2. Whenever two Apollo astronauts were on the surface of the Moon, a third
astronaut orbited the Moon. Assume the orbit to be circular and 100 km above the
surface of the Moon, where the acceleration due to gravity is 1.52 m/s2 . The radius
of the Moon is 1.70 × 106 m. Determine (a) the astronaut’s orbital speed and (b)
the period of the orbit.
3. In the Bohr model of the hydrogen atom, an electron moves in a circular path
around a proton. The speed of the electron is approximately 2.20 × 106 m/s. Find
(a) the force acting on the electron as it revolves in a circular orbit of radius 0.529
× 10−10 m and (b) the centripetal acceleration of the electron.
4. A curve in a road forms part of a horizontal circle. As a car goes around it at
constant speed 14.0 m/s, the total horizontal force on the driver has magnitude 130
N. What is the total horizontal force on the driver if the speed on the same curve is
18.0 m/s instead?
5. In a cyclotron (one type of particle accelerator), a deuteron (of mass 2.00 u)
reaches a final speed of 10.0% of the speed of light while moving in a circular
path of radius 0.480 m. What magnitude of magnetic force is required to maintain
the deuteron in a circular path?
6. A car initially traveling eastward turns north by traveling in a circular path at
uniform speed as shown in Figure P6.6. The length of the arc ABC is 235 m, and
the car completes the turn in 36.0 s. (a) What is the acceleration when the car is at
B located at an angle of 35.0°? Express your answer in terms of the unit vectors i
and j . Determine (b) the car’s average speed and (c) its average acceleration
during the 36.0-s interval.
Figure P6.6
7. A space station, in the form of a wheel 120 m in diameter, rotates to provide an
“artificial gravity” of 3.00 m/s2 for persons who walk around on the inner wall of
the outer rim. Find the rate of the wheel’s rotation in revolutions per minute that
will produce this effect.
8. Consider a conical pendulum (Fig. P6.8) with a bob of mass m = 80.0 kg on a
string of length L = 10.0 m that makes an angle of θ = 5.00° with the vertical.
Determine (a) the horizontal and vertical components of the force exerted by the
string on the pendulum and (b) the radial acceleration of the bob.
9. A coin placed 30.0 cm from the center of a rotating, horizontal turntable slips when
its speed is 50.0 cm/s. (a) What force causes the centripetal acceleration when the
coin is stationary relative to the turntable? (b) What is the coefficient of static
friction between coin and turntable?
Figure P6.8
10. Why is the following situation impossible? The object of mass m = 4.00 kg in
Figure P6.10 is attached to a vertical rod by two strings of length ℓ = 2.00 m. The
strings are attached to the rod at points a distance d = 3.00 m apart. The object
rotates in a horizontal circle at a constant speed of υ = 3.00 m/s, and the strings
remain taut. The rod rotates along with the object so that the strings do not wrap
onto the rod. What If? Could this situation be possible on another planet?
Figure P6.10
11. A crate of eggs is located in the middle of the flatbed of a pickup truck as the
truck negotiates a curve in the flat road. The curve may be regarded as an arc of a
circle of radius 35.0 m. If the coefficient of static friction between crate and truck
is 0.600, how fast can the truck be moving without the crate sliding?
Section 6.2 Nonuniform Circular Motion
12. A pail of water is rotated in a vertical circle of radius 1.00 m. (a) What two
external forces act on the water in the pail? (b) Which of the two forces is most
important in causing the water to move in a circle? (c) What is the pail’s minimum
speed at the top of the circle if no water is to spill out? (d) Assume the pail with
the speed found in part (c) were to suddenly disappear at the top of the circle.
Describe the subsequent motion of the water. Would it differ from the motion of a
projectile?
13. A hawk flies in a horizontal arc of radius 12.0 m at constant speed 4.00 m/s. (a)
Find its centripetal acceleration. (b) It continues to fly along the same horizontal
arc, but increases its speed at the rate of 1.20 m/s2 . Find the acceleration
(magnitude and direction) in this situation at the moment the hawk’s speed is 4.00
m/s.
14. A 40.0-kg child swings in a swing supported by two chains, each 3.00 m long.
The tension in each chain at the lowest point is 350 N. Find (a) the child’s speed at
the lowest point and (b) the force exerted by the seat on the child at the lowest
point. (Ignore the mass of the seat.)
15. A child of mass m swings in a swing supported by two chains, each of length R. If
the tension in each chain at the lowest point is T, find (a) the child’s speed at the
lowest point and (b) the force exerted by the seat on the child at the lowest point.
(Ignore the mass of the seat.)
16. A roller-coaster car (Fig. P6.16) has a mass of 500 kg when fully loaded with
passengers. The path of the coaster from its initial point shown in the figure to
point
B
involves only up-and-down motion (as seen by the riders), with no
motion to the left or right. (a) If the vehicle has a speed of 20.0 m/s at point
A
,
what is the force exerted by the track on the car at this point? (b) What is the
maximum speed the vehicle can have at point
Assume the roller-coaster tracks at points
A
B
and
and still remain on the track?
B
are parts of vertical circles of
radius r1 = 10.0 m and r2 = 15.0 m, respectively.
Figure P6.16 Problems 16 and 38.
17. A roller coaster at the Six Flags Great America amusement park in Gurnee,
Illinois, incorporates some clever design technology and some basic physics. Each
vertical loop, instead of being circular, is shaped like a teardrop (Fig. P6.17). The
cars ride on the inside of the loop at the top, and the speeds are fast enough to
ensure the cars remain on the track. The biggest loop is 40.0 m high. Suppose the
speed at the top of the loop is 13.0 m/s and the corresponding centripetal
acceleration of the riders is 2g. (a) What is the radius of the arc of the teardrop at
the top? (b) If the total mass of a car plus the riders is M, what force does the rail
exert on the car at the top? (c) Suppose the roller coaster had a circular loop of
radius 20.0 m. If the cars have the same speed, 13.0 m/s at the top, what is the
centripetal acceleration of the riders at the top? (d) Comment on the normal force
at the top in the situation described in part (c) and on the advantages of having
teardrop-shaped loops.
Figure P6.17
Frank Cezus/Getty Images
18. One end of a cord is fixed and a small 0.500-kg object is attached to the other end,
where it swings in a section of a vertical circle of radius 2.00 m as shown in
Figure P6.18. When θ = 20.0°, the speed of the object is 8.00 m/s. At this instant,
find (a) the tension in the string, (b) the tangential and radial components of
acceleration, and (c) the total acceleration. (d) Is your answer changed if the
object is swinging down toward its lowest point instead of swinging up? (e)
Explain your answer to part (d).
Figure P6.18
19. An adventurous archeologist (m = 85.0 kg) tries to cross a river by swinging from
a vine. The vine is 10.0 m long, and his speed at the bottom of the swing is 8.00
m/s. The archeologist doesn’t know that the vine has a breaking strength of 1 000
N. Does he make it across the river without falling in?
Section 6.3 Motion in Accelerated Frames
20. An object of mass m = 5.00 kg, attached to a spring scale, rests on a frictionless,
horizontal surface as shown in Figure P6.20. The spring scale, attached to the
front end of a boxcar, reads zero when the car is at rest. (a) Determine the
acceleration of the car if the spring scale has a constant reading of 18.0 N when
the car is in motion. (b) What constant reading will the spring scale show if the car
moves with constant velocity? Describe the forces on the object as observed (c) by
someone in the car and (d) by someone at rest outside the car.
Figure P6.20
21. An object of mass m = 0.500 kg is suspended from the ceiling of an accelerating
truck as shown in Figure P6.21. Taking a = 3.00 m/s2 , find (a) the angle θ that the
string makes with the vertical and (b) the tension T in the string.
Figure P6.21
22. A child lying on her back experiences 55.0 N tension in the muscles on both sides
of her neck when she raises her head to look past her toes. Later, sliding feet first
down a water slide at terminal speed 5.70 m/s and riding high on the outside wall
of a horizontal curve of radius 2.40 m, she raises her head again to look forward
past her toes. Find the tension in the muscles on both sides of her neck while she
is sliding.
23. A person stands on a scale in an elevator. As the elevator starts, the scale has a
constant reading of 591 N. As the elevator later stops, the scale reading is 391 N.
Assuming the magnitude of the acceleration is the same during starting and
stopping, determine (a) the weight of the person, (b) the person’s mass, and (c) the
acceleration of the elevator.
24. Review. A student, along with her backpack on the floor next to her, are in an
elevator that is accelerating upward with acceleration a. The student gives her
backpack a quick kick at t = 0, imparting to it speed υ and causing it to slide
across the elevator floor. At time t, the backpack hits the opposite wall a distance
L away from the student. Find the coefficient of kinetic friction μk between the
backpack and the elevator floor.
25. A small container of water is placed on a turntable inside a microwave oven, at a
radius of 12.0 cm from the center. The turntable rotates steadily, turning one
revolution in each 7.25 s. What angle does the water surface make with the
horizontal?
Section 6.4 Motion in the Presence of Resistive Forces
26. Review. (a) Estimate the terminal speed of a wooden sphere (density 0.830 g/cm3 )
falling through air, taking its radius as 8.00 cm and its drag coefficient as 0.500. (b)
From what height would a freely falling object reach this speed in the absence of
air resistance?
27. The mass of a sports car is 1 200 kg. The shape of the body is such that the
aerodynamic drag coefficient is 0.250 and the frontal area is 2.20 m2 . Ignoring all
other sources of friction, calculate the initial acceleration the car has if it has been
traveling at 100 km/h and is now shifted into neutral and allowed to coast.
28. A skydiver of mass 80.0 kg jumps from a slow-moving aircraft and reaches a
terminal speed of 50.0 m/s. (a) What is her acceleration when her speed is 30.0
m/s? What is the drag force on the skydiver when her speed is (b) 50.0 m/s and (c)
30.0 m/s?
29. Calculate the force required to pull a copper ball of radius 2.00 cm upward
through a fluid at the constant speed 9.00 cm/s. Take the drag force to be
proportional to the speed, with proportionality constant 0.950 kg/s. Ignore the
buoyant force.
30. A small piece of Styrofoam packing material is dropped from a height of 2.00 m
above the ground. Until it reaches terminal speed, the magnitude of its
acceleration is given by a = g − Bυ. After falling 0.500 m, the Styrofoam
effectively reaches terminal speed and then takes 5.00 s more to reach the ground.
(a) What is the value of the constant B? (b) What is the acceleration at t = 0? (c)
What is the acceleration when the speed is 0.150 m/s?
31. A small, spherical bead of mass 3.00 g is released from rest at t = 0 from a point
under the surface of a viscous liquid. The terminal speed is observed to be υT =
2.00 cm/s. Find (a) the value of the constant b that appears in Equation 6.2, (b) the
time t at which the bead reaches 0.632υT, and (c) the value of the resistive force
when the bead reaches terminal speed.
32. At major league baseball games, it is commonplace to flash on the scoreboard a
speed for each pitch. This speed is determined with a radar gun aimed by an
operator positioned behind home plate. The gun uses the Doppler shift of
microwaves reflected from the baseball, an effect we will study in Chapter 39.
The gun determines the speed at some particular point on the baseball’s path,
depending on when the operator pulls the trigger. Because the ball is subject to a
drag force due to air proportional to the square of its speed given by R = kmυ2 , it
slows as it travels 18.3 m toward the plate according to the formula υ = υie−kx.
Suppose the ball leaves the pitcher’s hand at 90.0 mi/h = 40.2 m/s. Ignore its
vertical motion. Use the calculation of R for baseballs from Example 6.11 to
determine the speed of the pitch when the ball crosses the plate.
33. Assume the resistive force acting on a speed skater is proportional to the square of
the skater’s speed υ and is given by f = −kmυ2 , where k is a constant and m is the
skater’s mass. The skater crosses the finish line of a straight-line race with speed
υi and then slows down by coasting on his skates. Show that the skater’s speed at
any time t after crossing the finish line is υ(t) = υi/(1 + ktυi).
34. Review. A window washer pulls a rubber squeegee ISO down a very tall vertical
window. The squeegee has mass 160 g and is mounted on the end of a light rod.
The coefficient of kinetic friction between the squeegee and the dry glass is 0.900.
The window washer presses it against the window with a force having a horizontal
component of 4.00 N. (a) If she pulls the squeegee down the window at constant
velocity, what vertical force component must she exert? (b) The window washer
increases the downward force component by 25.0%, while all other forces remain
the same. Find the squeegee’s acceleration in this situation. (c) The squeegee is
moved into a wet portion of the window, where its motion is resisted by a fluid
drag force R proportional to its velocity according to R = −20.0υ, where R is in
newtons and υ is in meters per second. Find the terminal velocity that the
squeegee approaches, assuming the window washer exerts the same force
described in part (b).
35. A motorboat cuts its engine when its speed is 10.0 m/s and then coasts to rest. The
equation describing the motion of the motorboat during this period is υ = υie−ct,
where υ is the speed at time t, υi is the initial speed at t = 0, and c is a constant. At
t = 20.0 s, the speed is 5.00 m/s. (a) Find the constant c. (b) What is the speed at t
= 40.0 s? (c) Differentiate the expression for υ(t) and thus show that the
acceleration of the boat is proportional to the speed at any time.
36. You can feel a force of air drag on your hand if you stretch your arm out of the
open window of a speeding car. Note: Do not endanger yourself. What is the order
of magnitude of this force? In your solution, state the quantities you measure or
estimate and their values.
Additional Problems
37. A car travels clockwise at constant speed around a circular section of a horizontal
road as shown in the aerial view of Figure P6.37. Find the directions of its
velocity and acceleration at (a) position
A
and (b) position
B
.
Figure P6.37
38. The mass of a roller-coaster car, including its passengers, is 500 kg. Its speed at
the bottom of the track in Figure P6.16 is 19 m/s. The radius of this section of the
track is r1 = 25 m. Find the force that a seat in the roller-coaster car exerts on a 50kg passenger at the lowest point.
39. A string under a tension of 50.0 N is used to whirl a rock in a horizontal circle of
radius 2.50 m at a speed of 20.4 m/s on a frictionless surface as shown in Figure
P6.39. As the string is pulled in, the speed of the rock increases. When the string
on the table is 1.00 m long and the speed of the rock is 51.0 m/s, the string breaks.
What is the breaking strength, in newtons, of the string?
Figure P6.39
40. Disturbed by speeding cars outside his workplace, Nobel laureate Arthur Holly
Compton designed a speed bump (called the “Holly hump”) and had it installed.
Suppose a 1 800-kg car passes over a hump in a roadway that follows the arc of a
circle of radius 20.4 m as shown in Figure P6.40. (a) If the car travels at 30.0 km/h,
what force does the road exert on the car as the car passes the highest point of the
hump? (b) What If? What is the maximum speed the car can have without losing
contact with the road as it passes this highest point?
Figure P6.40
Problems 40 and 41.
41. A car of mass m passes over a hump in a road that follows the arc of a circle of
radius R as shown in Figure P6.40. (a) If the car travels at a speed υ, what force
does the road exert on the car as the car passes the highest point of the hump? (b)
What If? What is the maximum speed the car can have without losing contact
with the road as it passes this highest point?
42. A child’s toy consists of a small wedge that has an acute angle θ (Fig. P6.42). The
sloping side of the wedge is frictionless, and an object of mass m on it remains at
constant height if the wedge is spun at a certain constant speed. The wedge is spun
by rotating, as an axis, a vertical rod that is firmly attached to the wedge at the
bottom end. Show that, when the object sits at rest at a point at distance L up
along the wedge, the speed of the object must be υ = (gL sin θ)1/2 .
Figure P6.42
43. A seaplane of total mass m lands on a lake with initial speed i i . The only
horizontal force on it is a resistive force on its pontoons from the water. The
resistive force is proportional to the velocity of the seaplane: R  b v . Newton’s
second law applied to the plane is b ˆi  m  d dt  ˆi From the fundamental
theorem of calculus, this differential equation implies that the speed changes
according to


i
d


b t
dt
m 0
(a) Carry out the integration to determine the speed of the seaplane as a function
of time. (b) Sketch a graph of the speed as a function of time. (c) Does the
seaplane come to a complete stop after a finite interval of time? (d) Does the
seaplane travel a finite distance in stopping?
44. An object of mass m1 = 4.00 kg is tied to an object of mass m 2 = 3.00 kg with
String 1 of length ℓ = 0.500 m. The combination is swung in a vertical circular
path on a second string, String 2, of length ℓ = 0.500 m. During the motion, the
two strings are collinear at all times as shown in Figure P6.44. At the top of its
motion, m2 is traveling at υ = 4.00 m/s. (a) What is the tension in String 1 at this
instant? (b) What is the tension in String 2 at this instant? (c) Which string will
break first if the combination is rotated faster and faster?
Figure P6.44
45. A ball of mass m = 0.275 kg swings in a vertical circular path on a string L =
0.850 m long as in Figure P6.45. (a) What are the forces acting on the ball at any
point on the path? (b) Draw force diagrams for the ball when it is at the bottom of
the circle and when it is at the top. (c) If its speed is 5.20 m/s at the top of the
circle, what is the tension in the string there? (d) If the string breaks when its
tension exceeds 22.5 N, what is the maximum speed the ball can have at the
bottom before that happens?
Figure P6.45
46. Why is the following situation impossible? A mischievous child goes to an
amusement park with his family. On one ride, after a severe scolding from his
mother, he slips out of his seat and climbs to the top of the ride’s structure, which
is shaped like a cone with its axis vertical and its sloped sides making an angle of
θ = 20.0° with the horizontal as shown in Figure P6.46. This part of the structure
rotates about the vertical central axis when the ride operates. The child sits on the
sloped surface at a point d = 5.32 m down the sloped side from the center of the
cone and pouts. The coefficient of static friction between the boy and the cone is
0.700. The ride operator does not notice that the child has slipped away from his
seat and so continues to operate the ride. As a result, the sitting, pouting boy
rotates in a circular path at a speed of 3.75 m/s.
Figure P6.46
47. (a) A luggage carousel at an airport has the form of a section of a large cone,
steadily rotating about its vertical axis. Its metallic surface slopes downward
toward the outside, making an angle of 20.0° with the horizontal. A piece of
luggage having mass 30.0 kg is placed on the carousel at a position 7.46 m
measured horizontally from the axis of rotation. The travel bag goes around once
in 38.0 s. Calculate the force of static friction exerted by the carousel on the bag.
(b) The drive motor is shifted to turn the carousel at a higher constant rate of
rotation, and the piece of luggage is bumped to another position, 7.94 m from the
axis of rotation. Now going around once in every 34.0 s, the bag is on the verge of
slipping down the sloped surface. Calculate the coefficient of static friction
between the bag and the carousel.
48. In a home laundry dryer, a cylindrical tub containing wet clothes is rotated
steadily about a horizontal axis as shown in Figure P6.48. So that the clothes will
dry uniformly, they are made to tumble. The rate of rotation of the smooth-walled
tub is chosen so that a small piece of cloth will lose contact with the tub when the
cloth is at an angle of θ = 68.0° above the horizontal. If the radius of the tub is r =
0.330 m, what rate of revolution is needed?
Figure P6.48
49. Interpret the graph in Figure 6.16(b), which describes the results for falling coffee
filters discussed in Example 6.10. Proceed as follows. (a) Find the slope of the
straight line, including its units. (b) From Equation 6.6, R 
1
D  A 2 , identify
2
the theoretical slope of a graph of resistive force versus squared speed. (c) Set the
experimental and theoretical slopes equal to each other and proceed to calculate
the drag coefficient of the filters. Model the cross-sectional area of the filters as
that of a circle of radius 10.5 cm and take the density of air to be 1.20 kg/m3 . (d)
Arbitrarily choose the eighth data point on the graph and find its vertical
separation from the line of best fit. Express this scatter as a percentage. (e) In a
short paragraph, state what the graph demonstrates and compare it with the
theoretical prediction. You will need to make reference to the quantities plotted on
the axes, to the shape of the graph line, to the data points, and to the results of
parts (c) and (d).
50. A basin surrounding a drain has the shape of a circular cone opening upward,
having everywhere an angle of 35.0° with the horizontal. A 25.0-g ice cube is set
sliding around the cone without friction in a horizontal circle of radius R. (a) Find
the speed the ice cube must have as a function of R. (b) Is any piece of data
unnecessary for the solution? Suppose R is made two times larger. (c) Will the
required speed increase, decrease, or stay constant? If it changes, by what factor?
(d) Will the time required for each revolution increase, decrease, or stay constant?
If it changes, by what factor? (e) Do the answers to parts (c) and (d) seem
contradictory? Explain.
51. A truck is moving with constant acceleration a up a hill that makes an angle 
with the horizontal as in Figure P6.51. A small sphere of mass m is suspended
from the ceiling of the truck by a light cord. If the pendulum makes a constant
angle θ with the perpendicular to the ceiling, what is a?
Figure P6.51
52. The pilot of an airplane executes a loop-the-loop maneuver in a vertical circle.
The speed of the airplane is 300 mi/h at the top of the loop and 450 mi/h at the
bottom, and the radius of the circle is 1 200 ft. (a) What is the pilot’s apparent
weight at the lowest point if his true weight is 160 lb? (b) What is his apparent
weight at the highest point? (c) What If? Describe how the pilot could experience
weightlessness if both the radius and the speed can be varied. Note: His apparent
weight is equal to the magnitude of the force exerted by the seat on his body.
53. Review. While learning to drive, you are in a 1 200-kg car moving at 20.0 m/s
across a large, vacant, level parking lot. Suddenly you realize you are heading
straight toward the brick sidewall of a large supermarket and are in danger of
running into it. The pavement can exert a maximum horizontal force of 7 000 N
on the car. (a) Explain why you should expect the force to have a well-defined
maximum value. (b) Suppose you apply the brakes and do not turn the steering
wheel. Find the minimum distance you must be from the wall to avoid a collision.
(c) If you do not brake but instead maintain constant speed and turn the steering
wheel, what is the minimum distance you must be from the wall to avoid a
collision? (d) Of the two methods in parts (b) and (c), which is better for avoiding
a collision? Or should you use both the brakes and the steering wheel, or neither?
Explain. (e) Does the conclusion in part (d) depend on the numerical values given
in this problem, or is it true in general? Explain.
54. A puck of mass m1 is tied to a string and allowed to revolve in a circle of radius R
on a frictionless, horizontal table. The other end of the string passes through a
small hole in the center of the table, and an object of mass m2 is tied to it (Fig.
P6.54). The suspended object remains in equilibrium while the puck on the
tabletop revolves. Find symbolic expressions for (a) the tension in the string, (b)
the radial force acting on the puck, and (c) the speed of the puck. (d) Qualitatively
describe what will happen in the motion of the puck if the value of m2 is increased
by placing a small additional load on the puck. (e) Qualitatively describe what will
happen in the motion of the puck if the value of m 2 is instead decreased by
removing a part of the hanging load.
Figure P6.54
55. Because the Earth rotates about its axis, a point on the equator experiences a
centripetal acceleration of 0.033 7 m/s2 , whereas a point at the poles experiences
no centripetal acceleration. If a person at the equator has a mass of 75.0 kg,
calculate (a) the gravitational force (true weight) on the person and (b) the normal
force (apparent weight) on the person. (c) Which force is greater? Assume the
Earth is a uniform sphere and take g = 9.800 m/s2 .
56. Galileo thought about whether acceleration should be defined as the rate of change
of velocity over time or as the rate of change in velocity over distance. He chose
the former, so let’s use the name “vroomosity” for the rate of change of velocity
over distance. For motion of a particle on a straight line with constant acceleration,
the equation υ = υi + at gives its velocity υ as a function of time. Similarly, for a
particle’s linear motion with constant vroomosity k, the equation υ = υi + kx gives
the velocity as a function of the position x if the particle’s speed is υi at x = 0. (a)
Find the law describing the total force acting on this object of mass m. (b)
Describe an example of such a motion or explain why it is unrealistic. Consider (c)
the possibility of k positive and (d) the possibility of k negative.
57. Figure P6.57 shows a photo of a swing ride at an amusement park. The structure
consists of a horizontal, rotating, circular platform of diameter D from which seats
of mass m are suspended at the end of massless chains of length d. When the
system rotates at constant speed, the chains swing outward and make an angle θ
with the vertical. Consider such a ride with the following parameters: D = 8.00 m,
d = 2.50 m, m = 10.0 kg, and θ = 28.0°. (a) What is the speed of each seat? (b)
Draw a diagram of forces acting on the combination of a seat and a 40.0-kg child
and (c) find the tension in the chain.
Figure P6.57
Stuart Gregory/Getty Images
58. Review. A piece of putty is initially located at point A on the rim of a grinding
wheel rotating at constant angular speed about a horizontal axis. The putty is
dislodged from point A when the diameter through A is horizontal. It then rises
vertically and returns to A at the instant the wheel completes one revolution. From
this information, we wish to find the speed υ of the putty when it leaves the wheel
and the force holding it to the wheel. (a) What analysis model is appropriate for
the motion of the putty as it rises and falls? (b) Use this model to find a symbolic
expression for the time interval between when the putty leaves point A and when it
arrives back at A, in terms of υ and g. (c) What is the appropriate analysis model
to describe point A on the wheel? (d) Find the period of the motion of point A in
terms of the tangential speed υ and the radius R of the wheel. (e) Set the time
interval from part (b) equal to the period from part (d) and solve for the speed υ of
the putty as it leaves the wheel. (f) If the mass of the putty is m, what is the
magnitude of the force that held it to the wheel before it was released?
59. An amusement park ride consists of a large vertical cylinder that spins about its
axis fast enough that any person inside is held up against the wall when the floor
drops away (Fig. P6.59). The coefficient of static friction between person and wall
is μs, and the radius of the cylinder is R. (a) Show that the maximum period of
revolution necessary to keep the person from falling is T = (4π2 Rμs/g)1/2 . (b) If the
rate of revolution of the cylinder is made to be somewhat larger, what happens to
the magnitude of each one of the forces acting on the person? What happens in the
motion of the person? (c) If the rate of revolution of the cylinder is instead made
to be somewhat smaller, what happens to the magnitude of each one of the forces
acting on the person? How does the motion of the person change?
Figure P6.59
60. Members of a skydiving club were given the following data to use in planning
their jumps. In the table, d is the distance fallen from rest by a skydiver in a “freefall stable spread position” versus the time of fall t. (a) Convert the distances in
feet into meters. (b) Graph d (in meters) versus t. (c) Determine the value of the
terminal speed υT by finding the slope of the straight portion of the curve. Use a
least-squares fit to determine this slope.
t (s) d (ft) t (s)
d (ft)
t (s)
d (ft)
0
0
7
652
14 1 831
1
16
8
808
15 2 005
2
62
9
971
16 2 179
3
138
10 1 138
17 2 353
4
242
11 1 309
18 2 527
5
366
12 1 483
19 2 701
6
504
13 1 657
20 2 875
61.A car rounds a banked curve as discussed in Example 6.4 and shown in Figure 6.5.
The radius of curvature of the road is R, the banking angle is θ, and the coefficient
of static friction is μs. (a) Determine the range of speeds the car can have without
slipping up or down the road. (b) Find the minimum value for μs such that the
minimum speed is zero.
62. In Example 6.5, we investigated the forces a child experiences on a Ferris wheel.
Assume the data in that example applies to this problem. What force (magnitude
and direction) does the seat exert on a 40.0-kg child when the child is halfway
between top and bottom?
63. A model airplane of mass 0.750 kg flies with a speed of 35.0 m/s in a horizontal
circle at the end of a 60.0-m-long control wire as shown in Figure P6.63a. The
forces exerted on the airplane are shown in Figure P6.63b: the tension in the
control wire, the gravitational force, and aerodynamic lift that acts at θ = 20.0°
inward from the vertical. Compute the tension in the wire, assuming it makes a
constant angle of θ = 20.0° with the horizontal.
Figure P6.63
64. A student builds and calibrates an accelerometer and uses it to determine the
speed of her car around a certain unbanked highway curve. The accelerometer is a
plumb bob with a protractor that she attaches to the roof of her car. A friend riding
in the car with the student observes that the plumb bob hangs at an angle of 15.0°
from the vertical when the car has a speed of 23.0 m/s. (a) What is the centripetal
acceleration of the car rounding the curve? (b) What is the radius of the curve? (c)
What is the speed of the car if the plumb bob deflection is 9.00° while rounding
the same curve?
Challenge Problems
65. A 9.00-kg object starting from rest falls through a viscous medium and
experiences a resistive force given by Equation 6.2. The object reaches one half its
terminal speed in 5.54 s. (a) Determine the terminal speed. (b) At what time is the
speed of the object three-fourths the terminal speed? (c) How far has the object
traveled in the first 5.54 s of motion?
66. For t < 0, an object of mass m experiences no force and moves in the positive x
direction with a constant speed υi. Beginning at t = 0, when the object passes
position x = 0, it experiences a net resistive force proportional to the square of its
speed: F net  mk 2 i , where k is a constant. The speed of the object after t = 0 is
given by υ = υi/(1 + kυit). (a) Find the position x of the object as a function of time.
(b) Find the object’s velocity as a function of position.
67. A golfer tees off from a location precisely at i  35.0 north latitude. He hits the
ball due south, with range 285 m. The ball's initial velocity is at 48.0° above the
horizontal. Suppose air resistance is negligible for the golf ball. (a) For how long
is the ball in flight? The cup is due south of the golfer’s location, and the golfer
would have a hole-in-one if the Earth were not rotating. The Earth’s rotation
makes the tee move in a circle of radius RE cos i   6.37 106 m  cos 35.0° as
shown in Figure P6.67. The tee completes one revolution each day. (b) Find the
eastward speed of the tee relative to the stars. The hole is also moving east, but it
is 285 m farther south and thus at a slightly lower latitude  f . Because the hole
moves in a slightly larger circle, its speed must be greater than that of the tee. (c)
By how much does the hole’s speed exceed that of the tee? During the time
interval the ball is in flight, it moves upward and downward as well as southward
with the projectile motion you studied in Chapter 4, but it also moves eastward
with the speed you found in part (b). The hole moves to the east at a faster speed,
however, pulling ahead of the ball with the relative speed you found in part (c). (d)
How far to the west of the hole does the ball land?
Figure P6.67
68. A single bead can slide with negligible friction on a stiff wire that has been bent
into a circular loop of radius 15.0 cm as shown in Figure P6.68. The circle is
always in a vertical plane and rotates steadily about its vertical diameter with a
period of 0.450 s. The position of the bead is described by the angle θ that the
radial line, from the center of the loop to the bead, makes with the vertical. (a) At
what angle up from the bottom of the circle can the bead stay motionless relative
to the turning circle? (b) What If? Repeat the problem, this time taking the period
of the circle’s rotation as 0.850 s. (c) Describe how the solution to part (b) is
different from the solution to part (a). (d) For any period or loop size, is there
always an angle at which the bead can stand still relative to the loop? (e) Are there
ever more than two angles? Arnold Arons suggested the idea for this problem.
Figure P6.68
69. The expression F = arυ + br2 υ2 gives the magnitude of the resistive force (in
newtons) exerted on a sphere of radius r (in meters) by a stream of air moving at
speed υ (in meters per second), where a and b are constants with appropriate SI
units. Their numerical values are a = 3.10 × 10−4 and b = 0.870. Using this
expression, find the terminal speed for water droplets falling under their own
weight in air, taking the following values for the drop radii: (a) 10.0 μm, (b) 100
μm, (c) 1.00 mm. For parts (a) and (c), you can obtain accurate answers without
solving a quadratic equation by considering which of the two contributions to the
air resistance is dominant and ignoring the lesser contribution.
70. Because of the Earth’s rotation, a plumb bob does not hang exactly along a line
directed to the center of the Earth. How much does the plumb bob deviate from a
radial line at 35.0° north latitude? Assume the Earth is spherical.